General relativity theory

Question

As I understand general relativity theory (please correct me if I'm wrong), time becomes dilated and space becomes compressed around mass, and this is responsible for gravity. I'm struggling with precisely how that results in gravitational force between masses. Can someone explain that to me (please keep any required math understandable to a lay person).

Answer 1

The theory of relativity consists of two by Albert Einstein formulated physical theories:

1. The special theory of relativity (1905):

The special relativity is a theory of spacetime structure and an evolution of the Newtonian notion of space and time which before that were considered to be absolut (e.g. no limitations to possible magnitudes of velocities were known before) and independent of each other under coordinate transformations (Galilei transformations). Special relativity is based on two postulations which contradict this physical space and time model:

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity).

2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.

Especially the second postulat would not be true, if the speed of light transformed as any other velocity under the Galilei coordinate transformations. For this postulates to be true, therefore the Galilei transformation needed to be altered to take into account the observation that there seemed to be no higher velocity then the speed of light (No physical object, message or field line can travel faster than the speed of light in a vacuum) and as a consequence the space time structure seemed to have a structure in which

1. time and space are not independent of each other under new to be found (Lorentz) coordinate transformations, and

2. observer which move with constant velocity to each other measure different values for the distance between two spatial points (length contraction) or different time differences (time dilatation) in their respective frame of reference (Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion).

The new defining feature of special relativity which takes this correlation between time and space into account is the introduction of the four vector concept. For example a space point is described by the four vector $x^{\mu}=\left(ct,\boldsymbol{x}\right)$, where $c$ is the speed of light and $\boldsymbol{x}$ the spatial three vector in Euclidean space $\mathbb{R}$. Every point in the four dimensional space time (called Minkowski space) is assigned such a four vector in such a way that $s^{2}=x_{\mu}x^{\mu}=g_{\mu\nu}x^{\mu}x^{\nu}=\left(ct\right)^{2}-\boldsymbol{x}^{2}$ remains invariant under Lorentz coordinate transformations. This is the defining feature of the notion of space and time in special relativity. $g_{\mu\nu}$ is the metric tensor by which the metric (e.g. the distance between different points in spacetime is defined) in the Minkowski space is definded. It is a constant quantity independent of $x^{\mu}$, which marks the Minkowski space as so called flat. This is the geometrical prescription of a flat space and is used throughout the special relativity in which gravitational forces between massed are not considered. This is where general theory of relativity sets in.

1. The general theory of relativity (1916):

General relativity is a theory of gravitation. An important principle is the equivalence principle, under which the states of accelerated motion (which cannot be accelerated to each other in special relativity) and being at rest in a gravitational field (standing on the earth´s surface) are not to be distinguished. What is different is that the free fall is inertial motion, a free falling object is falling since this is the way how objects move with no force acting on them, instead of this being due to the force of gravity as is the case in classical mechanics.

In classical mechanics one has a prefered special frames of reference defined by the movement in space and time, namely objects in free motion move along straight lines at constant speed (That means one needs due to Newton external forces to change this state of the object). But in general relativity the equivalence principle states a universality of free fall. The trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties, for example a person throwing a ball in an gravitational field will observe the same trajectory as the one throwing it in an accelarating object, provided that this accelaration induces the same relative force. This is summarized in the statement that the inertia mass $m_{a}$ and the gravitational mass $m_{g}$ are the same, $m_{g}=m_{a}$ (experimentally very precisely confirmed). This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This of course defines a new geometric structure of space and time, the notion of curved space time. The trajectories of free falling objects are geodesics obeying the geodesic equation and being the generalization of the notion of a straight line to curved space times.

The mathematical formulation of general relativity is based on differential geometry (especially of differential manifolds). For someone unexperienced in mathematics this is probably the highest challenge in tackling general relavitivity.

To answer your question, what you experience as a force in newtonian mechanics is in general relavitivity mathematical refomulated and encoded in a spacetime dependent metric $g_{\mu\nu}$, resulting in a curved spacetime described by a curvature tensor $R_{\mu\nu}$. These quantities are connected to the mass and energy content of a space time by the (by Einstein 1915 formulated) Einstein field equations:

$R_{\mu\nu}-\frac{1}{2}{R}g_{\mu\nu}=8\pi G T_{\mu\nu}$,

where $T_{\mu\nu}$ is the energy-momentum tensor containing the energy-mass distribution of a spacetime and $g_{\mu\nu}$ is the resulting $x$-dependent metric of the respective spacetime, which on tries to find (which is mathematically very difficult due to the nature of theses equations beeing a coupled system of differential equations for $g_{\mu\nu}$)

What you should keep in mind:

"Technically, general relativity is a theory of gravitation whose defining feature is its use of the Einstein field equations. The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially."

Masses induce a curved spacetime in which object move on geodesics which are not straight line in general any more, e.g. the famous experiment during a sun eclipse where it was confirmed that the light (the photons) of a far away star are deviated by the curved space around the heavy mass of the sun. This was a phenomenlogical affirmation of the model of gravitational forces in terms of a local metric and a curved spacetime created by masses in the spacetime, which would be without this mass, flat.

Answer 2

As Hasenet says in his / her answer, the biggest challenge you will face understanding general relativity (hereafter "GR") is the mathematics of differential geometry. I am not a Real Relativity-ist, but I do have a pretty good grip on the geometry side of things, so I’ll have a try to see whether I can help you along the lines of what I did for my daughter and friends at school.

You also might enjoy the very last chapter of the second volume of the Feynman Lectures on Physics. The chapter is called simply "Curved Space" and I would highly recommend you should read this.

For an intuitive understanding of GR you keep two things in mind:

1. John Wheeler's immortal summary: ""Mass tells space-time how to curve, and space-time tells mass how to move."
2. An accelerometer - as an instrument in a thought experiment – more on that soon – and, if you end up studying GR, I would go so far as to say teach yourself in some detail how real accelerometers work, especially the mass on spring ones and keep this working in mind as you’re doing the problems.

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Mass Tells Spacetime how to Bend – The Left Hand (Geometry Side) Side of the Einstein Field Equations

The essential form of the Einstein field equations (hereafter “EFE”) (the ones given by Hasenet without the cosmological constant which is the subject of much of cosmology at the moment) for your purposes in SI units is:

$\mathbf{G} = \frac{8\,\pi\,G}{c^4} \mathbf{T}$

(the scalar $G$ on the right being Newton's universal gravitational constant, $c$ the freespace lightspeed) and we could, if we liked, redefine our units so that they would read

$\mathbf{G} = \mathbf{T}$

Note that such a unit definition is not quite the one wontedly used by Real Relativity-ists.

Here $\mathbf{G} $ and $\mathbf{T}$ are $4\times4$ matrices – they are symmetric (left the same if you reflect the matrices in a line down their leading diagonal) so there seem to be 10 independent equations and they are matrix functions of the space and time co-ordinates, whose geometry is to be determined by solving the equations. The Einstein equations bend your mind a bit too, because what you are doing in solving them is discovering relationships (geometry) between the supposedly independent variables of the differential equations – this is an odd situation not found elsewhere in physics. Maxwell’s equations of electromagnetism, for example, have solutions specifying electromagnetic fields as functions of the independent space and time co-ordinates, and these co-ordinates truly are the “independent variables”. Certain further symmetries (Bianchi identities) lower the actual number of independent equations to 6. But just think of the above, matrix equation as a whole from now on. Also, there are four unspecified degrees of freedom to change co-ordinates, so its very complicated. But ultimately what you hope to get from the equations is the equations for the geodesics - the “straightest” possible lines one can have in curved geometry. These geodesics define the relationships between the spacetime co-ordinates to show us the geometry defined by the field equations. If the EFE were for two dimensional curved space, like a ball or other curved 2-surface, they would give you the equations for the straightest possible lines on the ball, and then you can think of pasting all these lines together in proper relationship to one another to build the manifold of spacetime, much as you weave a cloth from threads.

The left side of the equation $\mathbf{G}$, called the Einstein tensor, describes the geometry of space and time – how spacetime “bends”. This is an unvisualisable concept and so seems weird at first sight. But it is related to the logical extension of the intuitive curvature we intuitively understand by our everyday experience with balls. Mathematically the curvature tensor $\mathbf{R}$ (and the Einstein tensor is a simple function of this) simply tells us how we would fail if we tried to label space and time with “flat” co-ordinates, just as there is always distortion present on a flat map of a curved surface like the Earth’s surface. The jargon statement is "$\mathbf{R}$ measures the "block" to any attempt to integrate the tangent vector fields on the manifold to a Euclidean manifold". Another way of looking at this is, as discussed in Feynmann, is that a creature whose whole World were the surface of a ball and who couldn't see the ball from outside in 3-space could tell the ball were curved simply by drawing triangles – i.e. whose sides were the “straightest possible on the curved surface” and observing that their angles add up to more (on a positively curved surface like a sphere) than 180 degrees: indeed the discrepancy is proportional to the triangle’s area. (See the Wikipedia article on spherical trigonometry). Try imagining a triangle made up of a “straight line” (more generally called a geodesic or geodetic line – on the 2-sphere a great circle) drawn from the equator to one of the poles, a second geodesic from the pole to second point on the equator and then the last geodesic linking the two equatorial points. The sum of the angles is 180 degrees plus the difference between the longitudes of the two equatorial points. A second way you can detect the sphere’s curvature without going outside the sphere is by “parallel transporting” a vector around a closed curve on the surface. You should look up the Chinese “South Pointing Chariot” for a good visualisation of this idea – it is the idea of sliding a vector along a closed curve so that it is always a constant angle to the local geodesics. In general, when you get back to the beginning, the vector is pointing in a different direction from how it was when you began. The difference between the directions measures the curvature (the curvature tensor $\mathbf{R}$ defines the transformation between these two vectors in the manifold's "tangent space"). The extension of this parallel transport idea to higher dimensions is how mathematicians describe “curved” manifolds in higher dimensions. Try not to get too overwhelmed by not being able to really visualise curvature in higher dimensions – this mathematical idea of changes wrought by parallel transport around closed curves is “all there is too it” (it’s quite normal to feel quite frustrated and insecure with this idea when you begin thinking about it – after a while you understand that there is a degree of abstraction here beyond what we’ve evolved to perceive by our senses in our wonted, Euclidean 3-space, and you realise that the mathematics truly is “all there is”). A third and last way to notice your space is curved is to draw spheres and measure how much their surface area differs from the Euclidean geometry value of $4 \pi r^2$. For a visualisable 2-sphere, any circle (locus of all points at a constant geodetic-line distance from their centre) drawn on the sphere has a circumference (i.e. the measure of its boundary, just as the surface area is the measure of the sphere's boundary) that is less than $2\, \pi\,r$ where $r$ is the length of the geodesic joining the centre and any given point on the circumference. This third way of visualising curvature is the most relevant to the EFE as discussed in the Feynman lecture. It is actually less general than the parallel transport idea, and, accordingly, the Einstein tensor is a kind of “averaging” by a process called "contraction" of the more descriptive “Riemann Curvature Tensor” which casts the full parallel transport idea into symbols (one also subtracts some bits from the average to make the contracted tensor "traceless" - but that is an aside for now).

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Spacetime Tells Mass how to Move – The Right Hand (Physics Side) Side of the Einstein Field Equations

Now we look at the stress energy tensor $\mathbf{T}$. This is where much of the physics beyond pure geometry comes in, and I have a considerably weaker grasp of the ideas here, so hopefully some Real Relativity-ist can help me out here.

The stress energy tensor defines the spacetime distribution of the energy, sometimes appearing as matter i.e. stuff that fulfils the Pauli exclusion principle, sometimes appearing as “pure energy”, i.e. stuff that doesn’t fulfil the Pauli exclusion principle, like light, or radiation, especially in the early Universe. It’s all the same to the EFEs. The stress energy tensor also has certain symmetries and properties that ensure that things like local conservation of energy are true. Once you realise John Wheeler’s intuitive idea, most of the work in “discovering” the physical theory of gravitation is in understanding the form of $\mathbf{T}$.

But once you have the essential idea that this is the distribution of “stuff” in the World, you’ve essentially got it: the equation $\mathbf{G} = \mathbf{T}$ encodes John Wheeler’s wonderful sentence: the distribution of stuff in the world defines the spacetime geometry, which in turn defines how that stuff has to “move around” (distribute itself in space and time). The solutions of the EFE give us the geodesics in the geometry – equations of paths followed by small (in the sense of being too small to disturb $\mathbf{T}$) “test particles”.

Where does Force Come Into It?

GR does not deal with force directly. It simply tells you the geodesics and, by pasting these together, the geometry of the spacetime manifold. Gravity at its essence is the geometry of spacetime.

Force comes into the picture exactly as it always does in physics - through Newton’s second law. Now it’s time to bring out your imaginary accelerometer. A reference frame in “free fall” is one following the geodesics. A satellite orbiting the Earth is in “free fall” – it is following geodesics in the spacetime manifold defined by the stress energy tensor represented by the Earth’s presence. An accelerometer on the satellite would measure nought. Frames following geodesics generalise the inertial frame concept from special relativity.

However, if you want to deviate from a frame following geodesics (i.e. from frame of free fall, or a locally inertial frame), you need a force. This is Newton’s second law. You feel a force through your bottom on your seat or your shoes on the ground because the reference frame following the spacetime geodesic would be accelerating at $g$ metres per second through the surface of the Earth towards the Earth’s centre. However, you can’t do that – the matter in your body crunches up against the matter of the ground and your seat, and the latter pushes back on you so that you accelerate upwards relative to the free fall frame at $g$ metres per second. The force you feel on your bottom is, by Newton’s second law, simply $m\,g$. Read about the macabre fate of “spaghettification” that ultimately awaits all hapless astronauts falling into black holes. What is going on here – where are the forces coming from that spaghettify the hapless spacefarer? Inside the black hole, spacetime is highly “nonuniform” – the freefall frame varies swiftly between different but nearby points in spacetime. The freefall frame for your hand is significantly different from the freefall frame for your chest. So, by Newton II, if the spacefarer wants to keep their hand at the end of their arm, they need to apply a force to keep it attached. That force comes from tensions and compressions in their flesh and bones. Ultimately the tensile / compressive stresses needed to keep the different parts of the body deviating enough from their local freefall frames is too great for mortal flesh and bone, and the poor spacefarer drees his or her “spaghettified” weird.

One now understands an accelerometer to be: an instrument measuring the deviation of its frame of reference from the locally inertial (free fall) frame. I forgot to say that an entry level accelerometer won't do. You need one that also tells you the acceleration's direction - which sophisticated ones in aicraft for example always do. Also, given the spaghettification comments above, it has to be infinitely small and sensitive so it can probe neighbouring space time points separately and also infinitely light, so that it doesn't disturb the stress energy tensor.

A question I've seen several times on this site is whether acceleration is absolute in GR. One doesn't get a straight answer to this question, because it depends on what you mean by "acceleration". It is true that there is always a change of co-ordinates that can spirit the numerical "acceleration" away locally in any frame. But I think it is best conceptually to think of the accelerometer. Your frame of reference sitting on the Earth is not following a spacetime geodesic - it truly is accelerated relative to that geodesic and your accelerometer will measure this. In this standpoint, acceleration is absolute - it is the shape of the spacetime manifold and different reference frames passing through a point in spacetime in acceleration relative to each other at that point will get different readings on their accelerometers. From the accelerometer thought experiment standpoint, acceleration is indeed absolute.

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As well as Feynman, if you want to study further and grasp the GR project bit in your teeth, you might like to see the discussions of GR in Roger Penrose's "Road to Reality". I have been browsing a free to download book "A First Course in General Relativity" by Bernard Schutz. This one has an unusual and neat, thorough discussion at the end of the experimental measurement of gravitational waves and the technology and measurement procedures astronomers currently use to try to do this. The classic is "Gravitation" by Misner, Thorne and Wheeler. This is also very thorough, but a big project compared to the more succinct and modern Schutz and I think I would probably try Schutz first if I were studying GR for the first time. I've noticed on this site a book by Wald "General Relativity" discussed quite a bit but I have no experience of it other than a browse at bookshops. It can be cheaply bought in Kindle edition, which is seldom for technical books and this might be a plus.

Answer 3

I'm struggling with precisely how that results in gravitational force between masses.

It doesn't result in force between masses. In General Relativity, gravity is not a force, gravity is geodesic deviation.

The mathematics are formidable and, if one wants to understand precisely how it works, well, you'll just have to dig in and keep at it.

So, if you'll settle instead having a general notion of how it works, consider the following:

(1) free from any forces, the path of an object in spacetime (the object's world line) is a geodesic which loosely means that the path is, in some sense, as straight as it can be.

(2) if the spacetime is flat, the geodesics are straight lines and, importantly, two geodesics that are initially parallel will remain parallel.

(3) if the spacetime is curved, two initially parallel geodesics will, in general, not remain parallel.

Now, consider the surface of a globe and, in particular, the lines of longitude and latitude.

The lines of longitude are great circles and thus are geodesics of the surface of the globe.

Note that, at the equator, the lines of longitude are all perpendicular to the equator and thus, are initially parallel there.

However, as you move toward either pole, the lines of longitude converge, getting closer and closer together until meeting at a pole.

If these lines of longitude were the world lines of objects in some spacetime, you can see how it would appear that the objects "attract" one another and move closer and close until they collide.

Matter and energy curve spacetime such that the world lines, like the geodesic lines of longitude on a globe, do not remain parallel.

Again, if you wish to understand the particulars of the relationship between matter / energy and spacetime curvature, be prepared to devote quite a bit of time and effort. The other answers give a hint at the particulars.